GC 356 
.L8 B3 
Copy 1 



ADDITIONAL NOTES 



DISCUSSION OF TIDAL OBSERVATIONS 



MADE IN CONNECTION WITH THE 



COAST SURVEY AT CAT ISLAND, 



LOUISIANA. 



By Prof. A. D. BACHE, 

Superintendent U. S. Coast Survey. 



EXTRACTED FROM THE AMERICAN JOURNAL OF SCIENCE AND ARTS, VOL. XIV. 
SECOND SERIES. 



NEW FIAVEN: 

PRINTED BY B. L. HAMLEN, 

Printer to Yale College. 



1852. 



Q^.-^^^ 



1-^. 



I 



3^ 



In Exchange 

Peabody Inst, of Balto* 

June 14 1927 



ri 



"6 



Additional Notes of a Discussion of Tidal Observations made in 
connection with the Coast Survey at Cat Island, Louisiana / 
by Prof. A. D. Bache, Superintendent U. S. Coast Survey.* 



In my communication on the subject of thie tides at Cat Island, 
coast of Louisiana, at the New Haven meeting of the American 
Association,^ I showed that I had succeeded in decomposing the 
curves of rise and fall into a diurnal and semidiurnal curve, which 
were nearly curves of sines ; the diurnal curve having its maxi- 
mum approximately nine hours in advance of the first maximum 
of the semidiurnal curve, and the interference of these two waves 
producing the tidal wave as observed. The comparison of the 
curves deduced from the observations for three months, and the 
computed curves of sines, was shown to be satisfactory. This 
comparison, made as before by averages of periods of a week 
combined into one general mean, has now been extended to the 
whole year, as shown in the subjoined table. By increasing the 
maximum ordinate of the diurnal curve 0*02 of a foot, which 
will make the rise and fall agree more nearly with the average 
deduced from observation, we obtain, as shown in No. 2, a result- 
ing curve not differing in any ordinate more than a quarter of an 
inch from observation, and in which the positive and negative 
errors nearly balance, and the mean error deduced by summing 
the square of the errors is little more than one-eighth of an inch. 

* Read at the meeting of the American Association at Albany, and revised by 
the author, for publication in the American Journal of Science. 
•j- See this Jour., xii, 341. 



Prof. A. D. Bache on Tidal Observations. 



TABLE ]S"o. I. 



Showi) 


Z^rf/if 


comparison of diurnal and semidiurnal curves deduced from observations, 




m^/i curves of sines. 


Diagram No. 1. 






i 


No. 1. 


JNo. 1. 


iS 


No. 2. 








FROM OBSERVATJON. 


FROM CALCULATION. 


3 


FROM CA'Cl fATION. 


s 


(U 













i^i-- 


^ 


C CO 




> 

5 


"3 


3 


> 


"s 


"^ 


ii 


III! 






IS 


tt "a! 


^i- 


3 0," 




_ 


3 a; 


'-> (U 


^"S 


3^ «£:' 


3 9 


c 


t £ 


o 


C3 

E 


^ t 
ft. 


ft. 


CS 

c 

3 

c 
ft." 


— > 

in 


- t 

a 3 


c 
ft. 




11 


11 


1^ 
C 




ft. 


ft. 


ft. 


ft. 


ft. 


ft. 


ft. 





ooo 


000 


000 


C)-0O 


O-OC) 


000 


00 


o^oo 




o-oo 


O-oo 


I 


•17 


- -03 


•i4 


•i5 


- -04 


•I I 


•o3 


•i5 




•1 1 


•02 


2 


•3i 


- -06 


•25 


•28 


- -07 


•21 


•o3 


•29 


bJD 


•22 


•02 


3 


•44 


- •oS 


•36 


•4o 


- -08 


•32 


•o3 


•42 


c 


•34 


•02 


4 


•5i 


- -06 


•45 


•5o 


- ^07 


•43 


•02 


•5i 


"c 


■44 


•00 


5 


•56 


- -03 


•53 


■55 


- •04 


•5i 


•02 


57 


ili 


•53 


•00 


6 


•57 


- -oo 


•57 


•57 


— -oo 


•57 


•00 


•59 


^ 


•59 


- 02 


7 


•56 


^ -03 


•59 


•55 


+ '04 


•59 


•00 


•57 


CO 


-6j 


-•02 


8 


5i 


•06 


•57 


•5o 


•07 


•57 


•00 


•5i 


(D 


•58 


-•01 


9 


•44 


•08 


•52 


•4o 


•08 


•48 


•o3 


•42 


I 


•5o 


•02 


lO 


•3i 


•06 


•37 


•28 


•07 


•35 


•or 


•29 


U2 


•56 


•00 


II 


•17 


•o3 


•20 


•i5 


■04 


•19 


.01 


•i5 




•19 


•00 


12 


•00 


•00 


•00 


•00 


•00 


00 


•00 


•00 




•00 


•00 



























Nothing would be gained in closeness of representation of the 
result by displacing relatively the two tidal waves. It is only re- 
markable that in averages including the whole of the tides, even 
when most irregular, the results are so satisfactory. I have accord- 
ingly used the hypothesis of the representation of each wave by 
a curve of sines, deducing the maximum ordinate of computation 
from each observed ordinate. These laborious computations were 
made by Alexander S. Wadsworth, Jr., sub-assistant of the Coast 
Survey, and by Mr. P. B. Hooe. They give tables of heights of 
the diurnal and semidiurnal curve for each day of observation, 
which form the basis of the discussion of the heights. The next 
step after decomposing the curves of observation into diurnal and 
semidiurnal curves, is to discuss each separately to ascertain if 
they follow the laws deduced from them in regard to heights and 
times. 



1. Diurnal wave. 



Heights and times. 



If the diurnal curve is a curve of sines, then the ordinates found 
for each hour enable us to determine the value of the maximum 
or six-hour ordinate. Setting out from the mean line, then, we 
have for each day six determinations of the rise and fall above or 
below that line. Tables were computed from these, in which 
the daily curves were decomposed into their diurnal and semi- 
diurnal components. In making these tables, the very irregular 
tides have been in general omitted. These tables were arranged 
according to the moon's declination, beginning and ending with 
the days on which the declination was zero, determining the 
maximum ordinate of each day from zero of declination. As the 



Prof. A. D. Bache on Tidal Observations. 



irregular tides occur near the time of the moon's passing the 
equator, the averages of the heights about these times are de- 
duced from a less number of observations than the others, and 
are therefore less reliable. The following table gives the average 
heights, with the number of days from which they have been 
deduced. No advantage resulted from displacing the epoch of 
the moon's declination relatively to the day of highest tide. 

TABLE No. II. (Diagram 2.) 

Showing the value of the maximum ordinafes of the diurnal curve, on the several days 
from zero of dcc'inaiion of the moon to zero again, with the number of days from 
which the results are deduced. 



Days from zero 
of detlination. I 

No. of observa- 
tions. 


2 

0-33 


3 


4 


5 
0-59 


6 


7 


8 

0-77 


9 


10 

0-85 


II 


12 

0-70 


i3 


i4 








0^ 


Heights. ,0-33 


o-4i 


o65 


0-78 


0-8- 


0-77 


o-5g 


Nat. sin. 2 X 1 
moon's decliii'n 0-o5 


o-ii 


or; 


0-24 


0-4. 


0-46 


0-53 


0-58 


o-6o 


0-59 


0-54 


0-46 


0-37 


0-27 



The dependence of the height of the diurnal wave upon the 
moon's declination appears by comparing the lowest line of the 
table, containing the sine of twice the moon's declination, with 
the line next above it : it is also shown by the curves of Diagram 
No. 2. This agrees with Mr. Whewell's approximate formula 
for the diurnal inequality, namely, dh=C. sin 2d' ; in which dh is 
the difference in height of two consecutive high or low waters, 
C a constant, and S' the moon's declination. 

The variation of this same height with the sun's declination 
may be made at once apparent by classifying the heights for dif- 
ferent values of the sun's declination with the same declination 
of the moon. The following table contains the greatest heights 
of the diurnal curves during the several lunations of the year, 
with the values of the sun's declination and of the moon's dec- 
lination, grouped as described in the several columns. 

TABLE N'o. IIL (Diagram I^o. 3.) 
Showing the effect of change of sun's declination on height. 



Natural sine 2 sun's 


declination. 


Number of luna- 
tions in group. 


Natural sine 2 
moon's declination. 


Maximum ordinate 
diurnal curve. 


Greater than 


70^ 


5 


•572 


1-02 


70 to 


60 


6 


•577 


099 


60 to 


40 


6 


•565 


0-93 


4o to 


20 


5 


•53o 


094 


20 to 


00 


4 


•55o 


0-74 



The effect of the change of parallax of the moon may be 
shown satisfactorily by grouping the values of the heights at the 
greatest southern declination of the moon, and for the greatest 
northern declination, for the year ; comparing them for slightly 
varying declinations of the moon, for mean declinations of the 
sun, and for large variations of the parallax. The result is as 
shown in the following table, and in Diagram No. 4. 



Prof. A. D. Bache on Tidal Observations. 

TABLE No. IV. 
Showing the effect of change of moo7i^s parallax on height. 



Number 
of results. 


Mean sine 2 
moon's declina- 
tion both series. 

59-4 


M'n sine 'i sun's 

declination both 

series. 


M'n parallax 
correct, tor 
1st series. 


M'n parallax 
correct, for 
2d series. 


Mean height 
for lesser 
parallax. 


Mean height 

for greater 

parallax. 


i3i 


48-5 


5.-9 


65-9 


0-74 


0-88 



The parallax correction is taken as the cube of the parallax mul- 
tiplied by the sine of twice the moon's declination. 

These are the principal variable terms in the formula derived 
by Mr. Lubbock, from Bernouilli's theory of the tides, for the 
diurnal inequality, namely,* 

dh=zB[A . sin 2d . cos(v^ — qp)+ sin 2 5' . cos ip] ; 

in which dh is the difference in height of the morning and even- 
ing tide, B and A are constant coefficients, d' is the moon's de- 
clination and d the sun's ; i/ms a small variable to be added to the 
mean lunitidal interv^al to give the interval corresponding to the 
moon's age, and cp is the hour angle of the moon at the time of 
transit. The second term, introducing the parallax of the moon, 
would be 

p/3 

.sin2(^';t 



m . 



P3 



in which w is a constant coefficient, P is the mean parallax, and 
P^ the parallax at the time under consideration. 

In the application of this formula to the observations, the maxi- 
mum ordinates, found as before stated, were tabulated; and first 
the coefficients were deduced from the cases corresponding to the 
maximum of the sine of twice the moon's declination and to the 
minimum of the sun's, and vice versa, neglecting the small varia- 
tions due to cos(v^-g)) and cos xp. This gave the following values 
for the coefficients, and the two sets of equations derived con- 
formed with each other. 

TABLE ^o. V. 

Showing the value of coefficients deduced from, maximum sine twice moorHs declination 

and minimum of sun's, and vice versa ; neglecting variations due to 

cos(\^ — (p) and cos ^Jy. 





B. cos ^. 


B. A .cos(\Jy-(p). 


First six month?, 

Second six mouths, 

AVhole year, 


I -07 
i-oo 
I -04 


0-43 
0-39 

0-52 



As each day's results are referred to the mean level of the day, 
and the mean of the low and high waters is taken as giving the 
height of the diurnal tide, the constant from the mean level of 
the whole should not appear in the values. In beginning these 



* Transactions of the Royal Society of London, 1836, p. 223. 
f Lubbock's Elementary Treatise ou the Tides, London, 1839. 



Prof. A. D. Bache on Tidal Observations. 7 

researches, I did not suppose that small differences would come 
out of them such as have been deduced. The reference to the 
level of each day compensated in a degree for the effect of an 
entire raising or depressing of the water by the wind's action. 

The results promising success, the coefficients were deduced 
by the method of least squares for the first, and then for the sec- 
ond six months, and finally for the whole year. These laborious 
computations were made v/ith much skill by Mr. W. W. Gordon, 
of the Coast Survey. The result for the second six months, in 
reference to the coefficient of the term of the sun's declination, 
is discrepant from the final result ; but as the coefficients for the 
whole year were used, after endeavoring to trace the errors, if 
any, without immediate results, it was not pursued further. 

TABLE N-Q. VI 
Coefficients of cos {-^ - (p), deduced from the method of least squares. 





B.cos\^. B . A. cos(-4y-(p). 


First six months, 


i-oo 0-26 
0-90 o-6o 
096 0-24 


Second six months, 

Whole year, 





The sum of the positive and negative quantities balance, and 
rather better by the use of the coefficients from the first method, 
which differs chiefly in the coefficient of the sun's action. 

The coefficient of the first term of dh is Bx(A), and of the 
second term B; and it will be seen hereafter in discussing the 
semidiurnal tide, that (A) is 0-36, which, with 8 = 0-96, gives 
Bx(A) = 0-34. 

A set of tables was next made, containing the values of the two 
terms of the formula for each day. To these was subsequently 

p/3 

applied the small correction for the parallax from the term -— ; 

and the terms, being summed, were compared with the observed 
maximum ordinate, and the difference in the final column of the 
table showed the residual to be accounted for. 

For these tables I am indebted to Lieut. Trowbridge*- of the 
Corps of Engineers, assistant in the Coast Survey. The tabular 
quantities were also traced in curves, and then compared with the 
maximum ordinates. The positive and negative ditferetices are 
usually small, not exceeding in the average about (J- 12 of a foot, 
and are quite irregular. 

The irregularities apparent in the phenomena themselves in- 
duced me, in first commencing this investigation, to hope merely 
to be able to trace the phenomena generally ; but it now appears, 
from the character of the results obtained from the averages, that 
the theory may be followed much more closely by the results 
than I had at first supposed. 



8 



Prof. A. D. Bache on Tidal Observations. 



The accordance of observation and theory, after the corrections 
have been applied, is as good as the accidental errors of the sepa- 
rate results render necessary; as will be seen from the results for 
July given in the annexed table, and for July and part of August 
as given in Diagram No. 5 : but as the averages seemed to indi- 
cate that the residuals would show the laws of the phenomena, I 
discussed them further. 

TABLE N"o. VIL 

Showing the value of maximum ordinates of the diurnal curve, computed from the 
moon's declination and parallax, and from the sun's declination, compared with 
ordinates from observation, for the month of July. 





PAET OF A TABLE FOR THE YEAR. 




DAYS. 


Maximum ordin- ^ q» P 
ate. "'^^ • p 


-3. sin 2 5'. 0-26 


sin 2 5. 




July ] 


i-43 


66 


19 . 


60 


2 


•93 


59 
35 


19 


17 


3 


.96 


19 


43 


4 


•75 


33 


19 


26 


5 


•62 


23 


19 


17 


6 


.37 


08 


19 


10 


7 


•35 


l4 


^9 


12 


8 


•36 


i4 


\l 


o3 


9 


•32 


25 


09 


lO 


•52 


34 


18 


01 


II 


•65 


42 


18 


07 


12 


.75 


47 


•18 


i5 


i3 




54 


18 




i4 


•^8 


52 


18 


10 


i5 


•73 


53 


18 


02 


i6 


•62 


55 


18 — 


10 


17 


•89 


53 


17 


19 


i8 


48 


17 


29 


19 


•61 


4o 


17 


06 


20 


.52 


i5 


17 


20 


21 


.57 


00 


17 


4o 


22 


•4i 


02 


17 


22 


23 


•56 


33 


17 


07 


24 


•65 


45 


17 


o5 


25 


•61 


55 


16 — 


08 


26 


•77 


63 


16 


00 


27 


Z 


65 


16 


12 


28 


65 


16 


08 


29 


•90 


56 


16 


10 


3o 




48 


•16 


27 


3i 


•69 


37 


i5 


18 



In looking for an explanation of the irregularities to the terms 
(i// — (p) and v^, the residuals were classed according to the moon's 
age, and the averages taken for the separate hours. T'he result 
of these tables is given in that annexed, which shows the residual 
for each six months and for the year. I have introduced them 
for the half year, to show that the same law is deducihie, not- 
withstanding the irregularities of the individual results, from the 
observations for each six months. 



Prof, A. D. Bache on Tidal Observations. 



TABLE No. VIII. Diagram ]S"o. 6. 

Showing the residuals from the comparison of computed and observed ordinates of 
diurnal curves, classed according to the ages of the moon. 



Hours of moon's 
transit. 


RESIDUALS. 


First six months. 


Second six months. 


Mean. 


Oh 


•23 


•21 


•22 




^h 


•17 


•12 


•i3 




2i 


•i5 


•i5 


•i5 




3i 


•i5 


•12 


•i3 




4i 


•i6 


•00 


•08 




5^ 


•o8 


— o3 


■02 




64 


•o6 


— o3 


•01 




Ih 


•o8 


— ^02 


•o3 




8i 


•i3 


■04 


•08 




9i 


■12 


•12 


•12 




lOi 


•09 


•i4 


•II 




11.^ 


•19 


•i4 


•16 





These residuals, instead of following the law of cos (1/^-9), fol- 
low that of cos(2(// --2gD), or that of the semidiurnal curve. 

Before examining this result, which is shown in Diagram 6, I 
pass to the residual which results from carrying on the former 
table to 23J hours ; which was in fact the form of the table be- 
fore the development of the law of variation showed that the term 
for 12| hours belonged to OJ, instead of I IJ, with which it would 
agree if the law of cos(v^ — (jd) were followed. The following 
table contains the residuals in question, shown also in Diagram 
No. 7. 

TABLE No. IX. 

residuals after deducting those following law of change of 
cos (2\i/ - 2:p). 



Age of moon. 


Residuals. 


Residuals. 


Mean. 


hours. 


feet. 


hours. 


feet. 


Oh 


-•07 


23i 


- -01 




— •02 


22i 


— •01 


2^ 


•01 


2li 


•01 


3J 


•o3 


20h 


•o3 


Ah 


•00 


i9i 


•02 


5i 


•01 


i8.i 


•o4 


6^ 


•o5 


i7i 


•08 


ih 


•04 


16.^ 


•09 


8i 


•07 


iSh 


•o4 


9h 


•02 


lAh 


•00 


loh 


-•o3 


i^h 


•o3 


la 


•o3 


12I 


•06 






Me 


an -03 



The existence in the first residuals of the law belonging to the 
semidiurnal curve indicates that the separation of the two curves 
(diurnal and semidiurnal) is not complete, as indeed the hypothe- 
sis of a constant difference in time between the recurrence of the 
two maxima requires. Before undertaking to modify this hy- 
pothesis, 1 proceed to inquire whether these numbers would re- 



xo 



Prof. A. D, Bache on Tidal Observations. 



ceive modification from any otHer source. In examining the 
hypothesis that the component curves were curves of sines, a 
separation of the several hourly ordinates was necessary, and thus 
the four points at which the curves for twenty-four hours cross 
the hue of mean level were brought into consideration each day. 
Two of these points varied necessarily considerably in position, 
while the two twenty-four hours apart were regular. Having 
found that the curves of sines represent very nearly the observa- 
tion, the law thus obtained may be used in computing from all 
the hoiu'ly observations of the day the values of the maxitnnm 
ordinates for each ciu've ; forming the ordinates of the observed 
curve into groups containing resi)ectively the same positive and 
negative values of the ordinates of the diurnal curve, and again 
of the semidiurnal, arranging the groups for the consecutive 
twenty-four hours. It was soon apparent that the ordinates for 
the semidiurnal curve would in this way prove more considera- 
ble, in the average, than in the former mode of computation, and 
that the results would be more regular; that the ordinates of the 
diurnal curve would, on the average, be slightly diminished, and 
in general prove more regular. These revised tables have been 
prepared chiefly by Mr. W. W. Gordon and JMr. P. B. Hooe. They 
show on the average of the year a diminution of the maximum 
ordinates of the diurnal curve of 004 feet, and an increase of the 
maximum ordinates of the semidiurnal curve of 007 feet. 

Clavssifying the corrections according to the moon's age, though 
they are irregular, it is apparent that there were entangled in the 
values of the former computed maximum ordinates, heights which 
belonged to the semidiurnal curve. The table of correction for 
the two periods of six months, and for the year, is given below. 

TABLE 1^0. X. 

Showing the difference of maximum ordinates of diurnal curves, as computed hy the 
last method of groups, and hy that first applied. 



Time of moon's 
transit. 


Correction of 


maximum ordinates diurnal curve. 1 


First 6 months. 


Second 6 montiis. 


Mean of year. 


hours. 


feet. 


feet. 


feet. 


o.^ 


— •10 


-•06 


-•08 


4 


-|--o3 


-f-oS 


-|-o3 


2.i 


—08 


— •02 


-•o5 


3i 


—08 


-09 


-•08 


4i 


-•08 


-•09 


-•08 


5i 


-•o3 


-•o5 


- -04 


6^ 


— •02 


-•02 


— •02 




-•o5 


H-02 


-•01 


8^ 


-f-or 


•\--02 


+ •01 


9! 


— o5 


-oS 


-•o4 


loi 


—08 


-•o3 


-•o5 


Hi 


-•o3 


-•o4 


-•o3 



A consideration of the general formula for the height indicates 
a second correction. The height of high water, as given by the 



Prof. A. D. Bache on Tidal Observations. 



11 



formula is not the sum of the two greatest heights of the diurnal 
and semidiurnal tides. The hypothesis of the interference of 
the two waves makes the higli water the sum of two ordinates 
(neither of which is the maximum), depending upon the laws of 
increase and decrease of the curves respectively, and of the rela- 
tive position of the two ordiiiates. The correction due to this 
cause is readily found. The part of it which belongs to the 
diurnal curve will be the difference between D and D\ cos (/-E); 
where E, according to the hypothesis of the interference of the 
two waves, is 9 hours ; and t is the value for the maximum ordin- 
ate of the compound curve, namely (Proc. Amer. Assoc. Cam- 
bridge Meeting, page 289), 

4C 
cosec t - sec t— -^ ,, • 

This value of ^, containing C (the maximum ordinate of the semi- 
diurnal curve), shows that the quantity will vary with the time 
of the moon's transit, according to the half-monthly inequality of 
the height. Following the course which I have taken through- 
out this communication to give the resulting tables merelv, I sub- 

. . 40 

join the corrections thus derived from the tables for ^^ . from 

observation, the computed values of ^, and of D . cos (^ - E). The 
agreement of the general form of this correction with the theory 
is a new confirmation of the values of the quantities C and D, 
deduced from observation, which it contains. 



TABLE No. XL 

correction to height of the diurnal wave for difference of maximum ordinate, 
and of high water ordinate in cornpouTid curve. 





Correction to maximum 


Ti.ne of moon's transit. 


ordinate diurnal curve. 


hours. 


leet. 


oh 


—o3 


I^ 


— -oS 


2i 


— -03 


3^ 


—o4 


4h 


—o4 


5^ 


— •07 


6i 


— •08 


7h 


— •07 


H 


—06 


9h 


— -05 


loh 


— -oS 


llh 


—o4 



The correction furnished by the last two tables, and the cor- 
rected residual from the table, are given in Table No. 12 next 
following. 



12 



Prof. A. D. Bache on Tidal Observations. 



TABLE ]S"o. XII. 

Showivg residuals after correcting for new computations of ordinates, and difference 
between high water and maximum ordinates. 



Time of moon's 
transit. 


Correction of residual. 


Residual. 


Corrected residual. 


hours. 


fpct. 


feet. 


feet. 


o^ 


— -I r 


•22 


•I I 


li 


— -02 


•i3 


•II 


2i . 


— -08 


•i5 


•07 


3i 


— '12 


•i3 


•01 


4i 


-12 


•08 


-•04 


5^ 


— -11 


•02 


— •08 


6^ 


-lO 


•01 


— •08 


ih 


— •08 


•o3 


— •oS 


H 


— -oS 


•08 


•o3 


9f 


— 09 


•12 


•o3 


loh 


-lO 


•II 


•01 


"i 


— •07 


•16 


•09 
+ •21 
Mean. . . -017 



Comparing the residuals in this table with the uncorrected ones, 
we find their magnitude much decreased ; the average is now less 
than 0-02 of a foot : but the form of the series is, as before, that 
belonging to the semidiurnal curve, and is as well marked as 
when the quantities were more considerable. Diagram No. 6 
shows this fact ; containing the curve of residuals from Tables 8 
and 12, and of half-monthly inequality deduced from the obser- 
vations. This persistence in the form of the residuals affords 
the best evidence that the irregularities of the observations, and 
changes in the mode of computation, do not introduce errors of 
sufficient magnitude to mask the laws of the phenomena. I pro- 
pose therefore to modify the original hypothesis, so as if possible 
to obliterate this form in the residual. 

Some collateral questions have been examined in the course of 
this discussion, the results of which are interesting. One of these 
is the comparison of the maximum ordinates of the diurnal curve, 
corresponding to the moon's declination north and south. The 
average value of the sine of twice the moon's declination, and 
the corresponding average maximum ordinate for northern and 
southern declinations, are shown in the next table; from which 
it appears that if the values of sin 2(5' were equal, the heights 
would not differ appreciably. 

TABLE No. Xin. 

Showing the 7nean value of twice the moons declination, and the corresponding maxi- 
mum ordiyiates for northern and southern declinations. 



Sine 2y. 


Maximum ordinate. 


Sine 23'. 


Maximnm ordinate. 


•410 


•621 


•35i 
•410 


•538 
•53i 



Prof. A. D. Bache on Tidal Observations. 



13 



Another question was, whether the residuals, of which Table 
No. 7 shows a part, contained any portion which varied with the 
moon's dechnation. To test this, the residuals for six months were 
grouped accorduig to the declinations, with the following result. 

TABLE No. XIV. 

Containing the residuals after subtracting the terms containing the sine of twice the 
moon^s declination, and the sine of twice the suns declination, from the maximum 
ord'inates, grouped accordbig to the values of the sine of twice the moons declination. 





Average value of 


twice sine moon's declination. 1 


Groups, 


o to 20 
•]5t 
(33) 


20 io 35 

•i47 

(27) 


35 to 45 
•169 

(26) 


45 to 55 55 to 70 
•ii5 -267 

{M^ (37^ 


Average value, 

No. of observalion, . . 



The result indicates that there is no such term remaining in the 
residual. 

Another question was, as to whether changing the epoch would 
improve the results. Several attempts of this kind were made at 
different stages of the work, but without any marked advantage. 
The average result for the year, as shown by comparing the dates 
of occurrence of the greatest and least maxmium ordinate of the 
diurnal curve, and the greatest and least values of the term con- 
tainmg the moon's declination, is shown in the next table. The 
comparison is made in two different ways : first, by the date of 
the greatest vahie of the ordinate shown in the table of maximum 
ordinates; and secondly, by the date shown by the highest point 
of the curve, which was traced to represent the observations. 



TABLE No. XV. 

Showing results of comparison of dates of occ2crrence of the greatest and least maxi- 
mum ordinate of the diurnal curve, and the greatest and least value of term con- 
taining the moon's declination. 





DATE OF OCCUHRENCE 


— AVERAGE IN DAYS. 




Maxiirmm 

ordinate 

from table. 

i5-4 


Maxim ma 

ordinate 

from curve. 

i6-i 


'I'frm embracing 

sun and moon's 

declination. 

l6-o 


Minimum 

ordinate 

from table. 

i6-5 


Mmunum 

ordinate 

from curve. 

166 


Term containing 

Sim and moon's 

declination. 


160 



The times of occurrence of the maximum of the diurnal curve 
are, as I have already stated, connected by the hypothesis with 
those of the semidiurnal curve. The times deducible from the 
observations were so irregular, that I supposed it impracticable to 
do more than this. Notwithstanding all these irregularities, it 
turns out that the laws of the phenomena for the times are deduci- 
ble from the results. The average values follow those for the 
semidiurnal curve at the proper intervals. It will be practicable, 
therefore, to resume the examination of this part of the subject, 
which I accordnigly purpose to do. 



14 



Prof. A. D. Bache on Tidal Observations. 



2. Semidiurnal Curve. 

The results in relation to the semidiurnal curve have exceeded 
my anticipations. 'Vho. half-monthly inequality, hoth in height 
and time, is very well shown hy the maximum ordinates deduced ; 
though the greatest value of the height is only 022 feet, and the 
irregularities in the separate observed hitj;h waters fall upon hours 
instead of minutes. In the following table, the maximum ordin- 
ates obtained by the method of groups are used, and the small 
correction for the difference between maximum and high water 
ordinates is omitted. The latter contains time of moon's transit 
corresponding to observed height; and the height computed from 
the formula given by Mr. Lubbock as resulting from Bernouilli's 
theory, and the diiference between observation and theory. 

TABLE No. XVI. 
Shoivirig half-monthly inequality in height. 



Hours of moon's 
transit. 


Observed height. Compu 


ted height. 


o-c. 

Diff. of observed 
and computed. 


o^ 


•220 


223 


— oo3 


1 4 


•196 


206 


— -orG 


2I 


•199 


174 


•025 


3^ 


•147 


i3i 


•016 


^h 


•I 32 


087 


•045 


5i 


•074 


o56 


•018 


64 


•o47 


o56 


=ri 


74 


•074 


087 


84 


•it3 


]3i 


— •018 


9^ 


•i35 


174 


zt^ 


io4 


•i33 


206 


Hi 


•189 


223 


— ■o34 



The greatest difference between observed and computed heights 
is 0073, and the least difference 0003 ; and the mean, without 
regard to sine, is 0-026. Diagram No. 8 shows the observed and 
computed curves of half-monthly inequality of heights. The 
average interval corresponds to 2^1 3 5'^ of the moon's transit; 
which is therefore the zero point, or epoch of the half-monthly 
inequality in the interval. 

The interval corresponding to the moon's 





h m 




h 


m 


transit at 


3 30 


is 


11 


45 


" for 


9 30 


u 


13 


05 



Diff. is 1 20 
which, converted into arc, is 20°. 

Log tan 20° = log (A) = 9-56107; 
1 



(A) = 0-364; 
which is nearly the same as that obtained by Mr. Lubbock for Liv- 



X = 2-747; 



Prof. A. D. Bache on Tidal Observations. 



15 



erpool. The difference between the greatest and least heights is 

(0-220-0 047) = 0173 and E = ^^^ = 0238 : 

also the greatest height 0-220-=D + (E)x(l + A) = D-h '325; and 
D=-0 10. 



Since 



m' 



(0-07480; 



1 



M 



1 



???/H-M (A) 65-06' M 64 06 

For the half-monthly inequality of the intervals, we have 
(A)xsin2(p 0-364x si») 2(p 

tang 2 v^ = l_^(A)xcos'2^ "" lT0^364x~co72^ ' 
and in the heights, 

A^_0-10 + (E)x(A)xcos(2v/-2g)) + (E)cos2v' 
= - 0-104-0-087 X cos (2 xp -2(p) + 0-238 x cos 2 ip. 

The following table contains the half-monthly inequality of 
times deduced from the observations, and computed from the for- 
mula for tang 2 v^, and the comparison of observed and computed 
quantities. 

TABLE JSTo. XVII. 

Showing differences between the results obtained from the observations and from formula. 
Mean from observation 12h. 3-5m. 





^^ 


c. 


0. 


0- 


-c. 


(p 


From formula. 


From observation. 


+ 




h. m. 


h. m. 


h. m. 


h, m. 


m. 


m. 


3o 


o o8 


12 27 


12 3l 


o4 




I 3o 


23 


12 12 


12 3l 


19 




a 3o 


36 


n 59 


II 19 




4o 


3 3o 


42 


II 53 


II 45 




08 


4 3o 


38 


II 57 


12 o3 


06 




5 3o 


I? 


12 18 


12 24 


06 




6 3o 


17 


12 52 


12 38 




i4 


7 3o 


38 


i3 i3 


i3 09 




o4 


8 3o 


42 


i3 17 


i3 27 


10 




9 3o 


36 


i3 II 


i3 o5 




06 


10 3o 


23 


12 58 


i3 o5 


07 




II 3o 


o8 


12 43 


i3 o5 


22 




+74 


-72 



12h 35«i not being the exact mean of the observed times, the -{- 
and — differences do not balance exactly. 

Diagram No. 9 shows the observed and computed results. The 
greatest and least heights correspond with the average interval, as 
they should do by Bernouilli's theory. 

The average interval corresponds to 0^23°^ nearly, showing that 
transit E should be used instead of transit F. 



LIBRARY OF CONGRESS 



029 714 185 5 i 

I 



